大距离Heegaard分解的三穿孔球面和

设$M_i~(i=1,2)$是一个紧致可定向的三维流形, $F_i$是$M_i$边界上的一个不可压缩曲面, $M=M_{1}\cup_{f}M_{2}$, 其中$f$是$F_1$到$F_2$一个同胚,对于具有特定条件的相粘曲面$F_i$, 如果$M_i$具有一个Heegaard距离至少是$2(g(M_1)+g(M_2))+1$的Heegaard分解,则$g(M)=g(M_1)+g(M_2)$.     

划分围长至少为6 的平面图为有界大小的分支

图$G$的$(\mathcal{O}_{k_1}, \mathcal{O}_{k_2})$-划分是将$V(G)$划分成两个非空子集$V_{1}$和$V_{2}$, 使得$G[V_{1}]$和$G[V_{2}]$分别是分支的阶数至多$k_1$和$k_2$的图.在本文中,我们考虑了有围长限制的平面图的点集划分问题,使得每个部分导出一个具有有界大小分支的图.我们证明了每一个围长至少为6并且$i$-圈不与$j$-圈相交的平面图允许$(\mathcal{O}_{2}$, $\mathcal{O}_{3})$-划分,其中$i\in\{6,7,8\}$和$j\in\{6,7,8,9\}$.     

高度正则图的全着色

一个图$G$的全色数$\chi_T(G)$ 是对$G$的边和顶点着色的最小数, 使得相关联或相邻元素着不同色. 证明了如果$G$是正则图并且$d(G)\ge\cfrac{2}{3}|V(G)|+\cfrac{23}{6},$ 这里$d(G)$ 表示在$G$中顶点的度, 则$\chi_T(G)\leq d(G)+2$.     

一类复线性微分差分方程亚纯解的唯一性

本文主要研究一类复线性微分差分方程超越亚纯解的唯一性.特别地,假设$f(z)$为复线性微分差分方程: $W_{1}(z)f''(z+1)+W_{2}(z)f(z)=W_{3}(z)$的一个有穷级超越亚纯解,其中$W_{1}(z)$, $W_{2}(z)$, $W_{3}(z)$为增长级小于1的非零亚纯函数并且满足$W_{1}(z)+W_{2}(z)\not\equiv 0$.若$f(z)$与亚纯函数$g(z)$, $CM$分担0,1,$\infty$,则$f(z)\equiv g(z)$或$f(z)+g(z)\equiv f(z)g(z)$或$f^{2}(z)(g(z)-1)^2+g^{2}(z)(f(z)-1)^2=g(z)f(z)(g(z)f(z)-1)$或存在一个多项式$\varphi(z)=az+b_{0}$使得$f(z)=\frac{1-e^{\varphi(z)}}{e^{\varphi(z)}(e^{a_{0}-b_{0}}-1)}$与$g(z)=\frac{1-e^{\varphi(z)}}{1-e^{b_{0}-a_{0}}}$,其中$a(\neq 0)$, $a_{0}$ $b_{0}$均为常数且$a_{0}\neq b_{0}$.     

在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性

设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的     

不含某些子图的k连通图中的k可收缩边

最近Ando等证明了在一个$k$($k\geq 5$ 是一个整数) 连通图 $G$ 中,如果 $\delta(G)\geq k+1$, 并且 $G$ 中既不含 $K^{-}_{5}$,也不含 $5K_{1}+P_{3}$, 则$G$ 中含有一条 $k$ 可收缩边.对此进行了推广,证明了在一个$k$连通图$G$中,如果 $\delta(G)\geq k+1$,并且 $G$ 中既不含$K_{2}+(\lfloor\frac{k-1}{2}\rfloor K_{1}\cup P_{3})$,也不含 $tK_{1}+P_{3}$ ($k,t$都是整数,且$t\geq 3$),则当 $k\geq 4t-7$ 时, $G$ 中含有一条 $k$ 可收缩边.     

半无爪可迹图的度和条件

可迹图即为一个含有Hamilton路的图.令$N[v]=N(v)\cup\{v\}$, $J(u,v)=\{w\in N(u)\cap N(v):N(w)\subseteq N[u]\cup N[v]\}$.若图中任意距离为2的两点$u,v$满足$J(u,v)\neq \emptyset$,则称该图为半无爪图.令$\sigma_{k}(G)=\min\{\sum_{v\in S}d(v):S$为$G$中含有$k$个点的独立集\},其中$d(v)$表示图$G$中顶点$v$的度.本论文证明了若图$G$为一个阶数为$n$的连通半无爪图,且$\sigma_{3}(G)\geq {n-2}$,则图$G$为可迹图; 文中给出一个图例,说明上述结果中的界是下确界; 此外,我们证明了若图$G$为一个阶数为$n$的连通半无爪图,且$\sigma_{2}(G)\geq \frac{2({n-2})}{3}$,则该图为可迹图.     

图加一条边后的带宽和

令$BS(G,f)=\sum\limits_{uv\in E(G)}|f(u)-f(v)|$, 其中$f$为$V(G)\rightarrow\{1,2,\cdots,|V(G)|\}$的双射, 并称$BS(G)=\min\limits_{f}BS(G,f)$为图$G$的带宽和. 讨论顶点数为$n$的简单图$G$加上一条边$e\in\overline{E(G)}$后, 带宽和$BS(G+e)$与$BS(G)$的关系, 得其关系式$BS(G)+1\leq BS(G+e)\leq BS(G)+n-1$. 并证明此不等式中等号可取到, 即存在图$G_{1}$和$G_{2}$使得$BS(G_{1}+e)=BS(G_{1})+1$, $BS(G_{2}+e)=BS(G_{2})+n-1$.     

非齐性广义Morrey空间上双线性强奇异Calder\''{o}n-Zygmund算子及交换子

本文的主要建立非齐性度量测度空间上双线性强奇异积分算子$\widetilde{T}$及交换子$\widetilde{T}_{b_{1},b_{2}}$在广义Morrey空间$M^{u}_{p}(\mu)$上的有界性. 在假设Lebesgue可测函数$u, u_{1}, u_{2}\in\mathbb{W}_{\tau}$, $u_{1}u_{2}=u$,且$\tau\in(0,2)$. 证明了算子$\widetilde{T}$是从乘积空间$M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$到空间$M^{u}_{p}(\mu)$有界的, 也是从乘积空间$M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$到广义弱Morrey空间$WM^{u}_{p}(\mu)$有界的,其中$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$及$1相似文献   

一个关于平面图的K-(2,1)-全可选性的结果

设图$G$的一个列表分配为映射$L: V(G)\bigcup E(G)\rightarrow2^{N}$. 如果存在函数$c$使得对任意$x\in V(G)\cup E(G)$有$c(x)\in L(x)$满足当$uv\in E(G)$时, $|c(u)-c(v)|\geq1$, 当边$e_{1}$和$e_{2}$相邻时, $|c(e_{1})-c(e_{2})|\geq1$, 当点$v$和边$e$相关联时, $|c(v)-c(e)|\geq 2$, 则称图$G$为$L$-$(p,1)$-全可标号的. 如果对于任意一个满足$|L(x)|=k,x\in V(G)\cup E(G)$的列表分配$L$来说, $G$都是$L$-$(2,1)$-全可标号的, 则称$G$是 $k$-(2,1)-全可选的. 我们称使得$G$为$k$-$(2,1)$-全可选的最小的$k$为$G$的$(2,1)$-全选择数, 记作$C_{2,1}^{T}(G)$. 本文, 我们证明了若$G$是一个$\Delta(G)\geq 11$的平面图, 则$C_{2,1}^{T}(G)\leq\Delta+4$.     

沿着不连通曲面的Heegaard分解的不可稳定化的融合积

Let M be a 3-manifold, F= {F1 , F2 , . . . , Fn } be a collection of essential closed surfaces in M (for any i, j ∈ {1, ..., n}, ifi≠j, Fi is not parallel to Fj and Fi ∩Fj = φ) and0 M be a collection of components of M. Suppose M-UFi ∈FFi×(-1, 1) contains k components M1 , M2 , . . . , Mk . If each M i has a Heegaard splitting ViUSiWi with d(Si) > 4(g(M1 ) + ··· + g(Mk )), then any minimal Heegaard splitting of M relative to 0M is obtained by doing amalgamations and self-amalgamations from minimal Heegaard splittings or -stabilization of minimal Heegaard splittings of M1 , M2 , . . . , Mk .     

最大度与最小度相差不超过2的图的平衡judicious划分

图G的顶点集V(G)的一个二部划分V_1和V_2叫做平衡二部划分,如果||V_1|-|V_2||≤1成立.Bollobas和Scott猜想:每一个有m条边且最小度不小于2的图,都存在一个平衡二部划分V_1,V_2,使得max{e(V_1),e(V_2)}≤m/3,此处e(V_i)表示两顶点都在V_i(i=1,2)中的边的条数.他们证明了这个猜想对正则图(即△(G)=δ(G))成立.颜娟和许宝刚证明了每个(k,k-1)-双正则图(即△(G)-δ(G)≤1)存在一个平衡二部划分V_1,V_2,使得每一顶点集的导出子图包含大约m/4条边.这里把该结论推广到最大度和最小度相差不超过2的图G.     

WAVELET DECOMPOSITIONS IN $L^2(R^2)$

Let ${V_k}^{+\infinity}_{k=-\infinity}$ be a multiresolution analysis generated by a function $\phi(x)\in L^2(R^2)$. Under this multiresolution framework the key point for studying wavelet decompositions in $L^2(R^2)$ is to study the properties of Wo which is the orthogonal complement of $V_0$ in $V_1:V_1=V_0\oplus W_0$.In this paper the author studies the structure of W_0 and furthermore shows that a box spline of three directions can generate a wavelet decomposition of $L^2(R^2)$.     

The amalgamation of high distance Heegaard splittings is always efficient

Let M be a compact orientable manifold, and F be an essential closed surface which cuts M into two 3-manifolds M ₁ and M ₂. Let be a Heegaard splitting for i = 1, 2. We denote by d(S _i ) the distance of . If d(S ₁), d(S ₂) ≥ 2(g(M ₁) + g(M ₂) − g(F)), then M has a unique minimal Heegaard splitting up to isotopy, i.e. the amalgamation of and . Ruifeng Qiu is supported by NSFC(10625102).     

凸体的宽度不等式及应用

该文建立了关于单形宽度的杨路、张景中不等式的一个逆不等式. 作为凸体宽度不等式的应用,得到了凸体的截面和投影的一些估计式.     

Heegaard splittings 的曲面连通和

Suppose Mi = Vi ∪ Wi （i = 1,2） are Heegaard splittings. A homeomorphism f ： F1 → F2 produces an attached manifold M = M1 ∪F1=F2 M2, where Fi ∪→ δ_Wi. In this paper we define a surface sum of Heegaard splittings induced from the Heegaard splittings of M1 and M2, and give a sufficient condition when the surface sum of Heegaard splitting is stabilized. We also give examples showing that the surface sum of Heegaard splittings can be unstabilized. This indicates that the surface sum of Heegaard splittings and the amalgamation of Heegaard splittings can give different Heegaard structures.     

Proper Harmonic Maps from Hyperbolic Riemann Surfaces into the Euclidean Plane

Let Σ be a compact Riemann surface and D ₁ , . . . , D _n a finite number of pairwise disjoint closed disks of Σ. We prove the existence of a proper harmonic map into the Euclidean plane from a hyperbolic domain Ω containing ${\Sigma \backslash \cup_{j=1}^n D_j}$ and of its topological type. Here, Ω can be chosen as close as necessary to ${\Sigma \backslash \cup_{j=1}^n D_j}$ .